Logical Consistency Ontology

FLOW ALGEBRA
TITLE: Logical Consistency Ontology
DOMAIN: Discrete Mathematics / Logic
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/logic/logical-consistency-ontology-2.html
VERSION: 1.0
STATUS: Mathematical reference model; source page not independently verified

1. PURPOSE
Represent logical consistency as the absence of a derivable contradiction in a formal theory, distinguishing syntactic consistency from semantic satisfiability and recording the assumptions required to connect them.

2. TYPED ENTITIES
Language : FormalLanguage
Formula : WellFormedFormula(Language)
Theory : Set[Formula]
Interpretation : ModelCandidate(Language)
ProofSystem : DeductiveCalculus(Language)
Derivation : FiniteProofSequence
Contradiction : Formula AND Negation(Formula)
ConsistencyStatus : {CONSISTENT, INCONSISTENT, UNKNOWN}
Evidence : {Derivation, Model, SolverCertificate, Counterexample}

3. CORE RELATIONS
MEMBER_OF(phi, T) : phi in T
DERIVES(T, phi, C) : T |-C phi
SATISFIES(M, phi) : M |= phi
MODEL_OF(M, T) : forall phi in T, M |= phi
CONTRADICTS(phi, psi) : psi = NOT(phi)
SYNTACTICALLY_CONSISTENT(T, C) : NOT DERIVES(T, bottom, C)
SEMANTICALLY_SATISFIABLE(T) : exists M, MODEL_OF(M, T)

4. FORMAL OPERATIONS
CHECK_CONSISTENCY(T, C) -> ConsistencyStatus + Evidence
ADD_AXIOM(T, phi) -> T union {phi}
REMOVE_AXIOM(T, phi) -> T minus {phi}
EXTRACT_UNSAT_CORE(T) -> subset U of T, if supported
BUILD_MODEL(T) -> Model | UNKNOWN
FIND_REFUTATION(T, C) -> Derivation | UNKNOWN

5. MATHEMATICAL LAWS AND CONDITIONS
L1: If T |-C bottom, then T is syntactically inconsistent relative to C.
L2: If T has a model and C is sound, T cannot derive bottom.
L3: If C is sound and complete for the selected semantics, syntactic consistency and satisfiability coincide.
L4: If T is consistent, every subset S of T is consistent (same proof system).
L5: If T is inconsistent, every superset U of T is inconsistent (monotonic derivability).
L6: T union {phi} can become inconsistent even if T is consistent.
L7: Classical explosive logics validate: from contradiction, any formula is derivable; do not generalise to paraconsistent logics.
L8: For classical propositional logic, T is satisfiable iff no contradiction is derivable in a sound, complete calculus.
L9: In first-order classical logic, the compactness theorem yields: T satisfiable iff every finite subset is satisfiable.
L10: Consistency is always relative to a specified language, proof system and underlying logic.

6. INFERENCE RULES
R1 REFUTATION: DERIVES(T, bottom, C) => INCONSISTENT(T, C).
R2 MODEL_WITNESS: MODEL_OF(M, T) AND SOUND(C) => CONSISTENT(T, C).
R3 SUBSET_PRESERVATION: CONSISTENT(T, C) AND S subset T => CONSISTENT(S, C).
R4 SUPERSET_REFUTATION: INCONSISTENT(T, C) AND T subset U => INCONSISTENT(U, C).
R5 AXIOM_CONFLICT: DERIVES(T, phi, C) AND DERIVES(T, NOT(phi), C) AND C supports contradiction introduction => INCONSISTENT(T, C).
R6 NO_PROOF_FOUND: absence of a discovered refutation => UNKNOWN, not CONSISTENT.

7. SEMANTIC RELATIONSHIPS
LogicalConsistency --property_of--> Theory
LogicalConsistency --depends_on--> ProofSystem
LogicalConsistency --contrasts_with--> LogicalInconsistency
LogicalConsistency --related_to--> Satisfiability
LogicalConsistency --tested_by--> RefutationSearch
LogicalConsistency --supported_by--> ModelWitness
LogicalConsistency --used_in--> FormalVerification
LogicalConsistency --used_in--> KnowledgeBaseValidation

8. SEMANTIC_EQUIVALENCE
'contradiction-free' ~ 'syntactically consistent' only when contradiction-free means non-derivability of bottom in the named calculus.
'has a model' ~ 'satisfiable' under the chosen model-theoretic semantics.
'consistent' != 'complete'; a consistent theory need not decide every sentence.
'consistent' != 'true'; a consistent theory may have unintended or factually false models.

9. VALIDATION
V1: Check all formulas are well formed in the declared language.
V2: Record the selected logic and proof calculus.
V3: Verify every derivation step against an authorised rule.
V4: Verify model witnesses against every axiom of T.
V5: Do not infer consistency merely from solver timeout or failure to find proof.
V6: Label undecidable or resource-limited results UNKNOWN.
V7: Preserve provenance of assumptions, proofs and model certificates.
V8: Check that semantic/syntactic equivalence claims carry soundness and completeness assumptions.

10. CONFIDENCE
CONFIDENCE.status : {PROVEN, MODEL_VERIFIED, HEURISTIC, UNKNOWN}
CONFIDENCE.score : Real[0,1] | null
CONFIDENCE.basis : {proof_certificate, checked_model, solver_report, human_review}
Rule: confidence scores describe evidence quality; they do not replace mathematical proof.

11. FLOW_HASH
FLOW_HASH.algorithm : SHA-256
FLOW_HASH.input : canonical_utf8(language, logic, calculus, theory, rules, version)
FLOW_HASH.value : COMPUTE_AT_INGESTION
Rule: recompute on semantic changes; never use a fabricated digest.

12. UPDATE_MODEL
UPDATE_MODEL.version : 1.0
UPDATE_MODEL.trigger : {axiom_change, proof_rule_change, logic_change, semantic_revision}
UPDATE_MODEL.actions : {revalidate_formulas, rerun_checks, invalidate_stale_evidence, recompute_hash}
UPDATE_MODEL.history : append_only

13. AI_CONSUMPTION
AI_CONSUMPTION.type : typed_formal_logic_ontology
AI_CONSUMPTION.tasks : {consistency_checking, contradiction_detection, theorem_proving, knowledge_graph_auditing, formal_verification}
AI_CONSUMPTION.required_context : {logic, language, theory, proof_system, assumptions}
AI_CONSUMPTION.outputs : {status, evidence, provenance, confidence, limitations}
AI_CONSUMPTION.safety : Never treat UNKNOWN as CONSISTENT; distinguish proof from heuristic evidence.

14. EXAMPLE
T = {P -> Q, P, NOT Q}
From P -> Q and P infer Q (modus ponens).
With NOT Q, derive contradiction.
CHECK_CONSISTENCY(T, classical_propositional_logic) = INCONSISTENT.
T2 = {P -> Q, P}
Model M with P=true, Q=true satisfies T2.
CHECK_CONSISTENCY(T2, classical_propositional_logic) = CONSISTENT, with model witness.

AUTHORSHIP
Mark Higgins — Sky (Recursive AI)
Kireth Research | https://kirethresearch.org